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برگزاری کارگاهی تحت عنوان Ergodic properties of non-dissipative actions


مسولان برگزاری کارگاه: دکتر امین طالبی و دکتر حسام رجب زاده از IPM  تهران

Abstract. One of the main subjects in dynamical systems is to study the ergodic behaviors of typical orbits from viewpoint of a reference probability measure. When the reference measure is invariant, the classical machinery of ergodic theory provides lots of additional information about the system, such as recurrence and the existence of time averages for typical points. However, the reference measure may not be invariant under the iterations of the system. In general, the systems with a reference probability measure are divided into two categories: dissipative and non-dissipative systems. The latter contains the measure-preserving ones. This minicourse intends to introduce the literature for answering the question that to what extent the results in classical ergodic theory hold for all maps in the non-dissipative category. We will start with a short introduction of this category in an abstract setting, presenting different examples and reviewing their basic properties. In particular, a generalization of the notion of ergodicity for non-dissipative systems will be introduced. For the rest, we will restrict ourselves to smooth systems. We will investigate the stability of the generalized notion of ergodicity and propose a local mechanism for providing stably ergodic actions. Finally, we will talk about the convergence of time averages of typical points under iteration of the system and give an example to show the classical Birkhoff ergodic theorem fails in this setting, even if the system is ergodic.

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